Destroying Densest Subgraphs is Hard

TL;DR

This work investigates the robustness of densest subgraphs under graph modifications by introducing Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion. The authors establish a nuanced complexity landscape: both problems are polynomial-time solvable on trees and cliques, while becoming NP-hard on planar bipartite and split graphs; parameterized results show fixed-parameter tractability w.r.t. vertex-cover number but W[1]-hardness w.r.t. the solution size, with the edge-deletion variant also W[1]-hard when combined with the feedback-edge number. The analysis connects these problems to classic graph-modification problems (e.g., Feedback Vertex/Edge Set, Vertex Cover, and X3C), and provides ILP-based FPT algorithms alongside strong hardness results, including W[1]-hardness for treewidth in certain reductions. The findings illuminate the delicate balance between tractability and hardness driven by target density and graph structure, and point to open questions on treewidth, other parameters, and broader density regimes. Overall, the paper advances the parameterized complexity understanding of modifying graphs to bound densest subgraphs with implications for network analysis applications.

Abstract

We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion: Given a graph $G$, a budget $k$ and a target density $τ_ρ$, are there $k$ edges ($k$ vertices) whose removal from $G$ results in a graph where the densest subgraph has density at most $τ_ρ$? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that Bounded-Density Edge Deletion is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs.

Figures (2)

• Figure 1: The computational complexity and special cases of Bounded-Density Edge Deletion for specific values of the target density $\tau_\rho$, see \ref{['sec:poly-time-edge-del', 'sec:np-hardness-edge-del']} for the details. Green (hatched) boxes indicate polynomial-time solvable cases while red (solid) boxes denote NP-hard cases. The $c$ in $1+1/c$ can be any constant larger than 24. The complexity for larger values of $\tau_\rho$ remains open.
• Figure 4: An example of a graph $G$ and a densest subgraph $H'$ in $S(G)$ which is not a subdivision of a subgraph of $G$.

Theorems & Definitions (27)

• Lemma 1
• Lemma 2
• Proposition 6
• Theorem 7
• Theorem 8
• Theorem 9: RV1986
• Theorem 10
• Theorem 11
• Theorem 12
• Corollary 13